ooo3-4916/86 s7.50. Copyright ... can be written as 6s = 0. where the action ...... J. MARSDEN AND R. WALTON, Comm. Math. Ph.vs. 99 (1985), 319-345. 4. A. 0.
ANNALS
OF PHYSICS
169, 2947
Covariant
(1986)
Poisson
Brackets for Classical
Fields
J. E. MARSDEN AND R. MONTGOMERY Department
of Mathematics, Universiry of California, Berkeley, California 94720
P. J. MORRISON Deparfmem of Physics and Institute University qf Texas, Ausrin,
jbr Fusion Studies, Texas 78712
AND
W. B. THOMPSON Department of Physics, University San Diego, La Jolla, California
of California. 92093
Received January 8, 1985
Poisson brackets that are spacetime covariant are presented for a variety of relativistic field theories. These theories include electromagnetism, general relativity, and general relativistic fluids and plasmas in Eulerian representation. The examples presented suggest the development of a general theory; the beginnings of such a theory are presented. Our covariant bracket formalism provides a general setting for. amongst other things, clarifying the transition from the covariant formalism to the dynamical 3 + 1 Hamiltonian formalism of Dirac and Arnowitt, Deser, and Misner. We illustrate the relevant procedures with electromagnetism. 4? 1986 Academic
Press. Inc.
1. INTRODUCTION The purpose of this paper is to show how to write the equations of some specific general relativistic field theories in covariant Poisson bracket form. Our approach is to proceed from explicit examples to some speculations on the structure of the underlying mathematical theory. For each of the examples, the field equations will be shown to be equivalent to equations of the form
where F is an arbitrary function of the fields and S is an action integral. theories considered fall into two categories:
The
A. Purefields, typified by gauge fields, where F and S in (1.1) are functions of the basic field variables 4” and their conjugate momenta ~5. 29 ooo3-4916/86 s7.50 Copyright 0 1986 by Academic Press. Inc. All rights of reproduction in any form reserved.
30
MARSDEN
ET AL.
B. Media fields, such as those describing relativistic fluids and plasmas in Eulerian representation, where F and S are just functions of the basic fields (without the addition of conjugate momenta). In either case, the formalism has these features: 1. It involves an integration over both space and time. 2. Equation (1.1) satisfies the usual properties of Poisson brackets, such as Jacobi’s identity, so the space of fields forms a Poisson manifold (see Dirac [9]). 3. S is the action (Lagrangian) suitably expressed as a function of the field variables. For pure fields, the bracket contains a spacetime vector field V’ which would correspond to the choice of a slicing, were a 3 + 1 Dirac-ADM decomposition performed (see Fischer and Marsden [ 111 and Isenberg and Nester [ 191 for reviews). For media fields, the bracket is a covariant extension of brackets of Lie-Poisson type that are now common for 3 + 1 relativistic and non-relativistic media fields (Iwinski and Turski [20], Morrison [37], Marsden and Weinstein [34], etc.). We reiterate: the aim of this paper is purely observational. We simply observe that many relativistic field theories can be written in covariant Poisson bracket form. The basic mathematical underpinnings of the present work are not claimed to be worked out. To complete the basic theory, one should tie up the present results with the multisymplectic approach (see for example, Kijowski and Tulczyjew [ZS]). It is anticipated that the covariant Poisson bracket formalism will be useful for calculation. In this direction, Kaufman and Holm [22] have used a covariant single particle bracket (due to Ignatiev) with success. The present formalism would be interesting to pursue along these and other lines. To motivate some of our results, let us first consider the simple case of particle mechanics. We recall that the canonical Hamilton equations
aff "'=j$
. ai pi=-q'
(1.2)
can be written as 6s = 0. where the action
SCYI =J (Pi4’-ff(q,P))
dt
(1.3)
is regarded as a functional on r, the space of paths y(t) = (q(t), p(t)) in phase space with appropriate boundary conditions (see, e.g., Arnold [2, p. 2431). Let us rewrite this variational principle in terms of a Poisson bracket on r. For functionals F and G of paths y, set
(1.4)
COVARIANT
where the functional
POISSON
BRACKETS
FOR CLASSICAL
FIELDS
31
derivatives are defined by
for variations 6y vanishing at the endpoints of y. It is straightforward &S[r] = 0; i.e., y solves Hamilton’s equation if and only if {F, S)(y)=0
to check that (1.6)
for all functionals F. It is this variational principle for Hamilton’s equations that we shall generalize and apply to field theory. The covariant theory does not, of course, single out a time direction; rather space and time occur on equal footing, as will be seen below. For general covariant brackets, the operation FH (F, S> may be viewed as a variation of the action S along a direction in function space determined by F. In this sense, (1.1) can be viewed as a reformulation of the conventional variational approach to field theory. However, it is also a generalization and unifying principle, for conventional field theories treat electromagnetism and fluids, for example, in a rather different way. Field theories of the traditional Euler-Lagrange form have been analyzed by symplectic methods in a fairly well-developed way. See, for example, Chernoff and Marsden [7], Dedecker [8], Abraham and Marsden [l], Kijowski and Tulczyjew [25] and Gimmsy [14] and references therein. However, a number of important field theories do not lit this mold, just as rigid body equations in body representation, i.e., in terms of the body angular momenta, admit a simple Poisson description, but not a symplectic one (since, for example, there are three equations); see for example, Sudarshan and Mukunda [41] and Holmes and Marsden [ 181 for these descriptions. There are also Poisson bracket formulations of the equations of fluids and plasmas in Eulerian description, as is now well known. For reviews, see Morrison [38], Holm and Kupershmidt [17], Marsden, Weinstein, Ratiu, Schmid and Spencer [35], Marsden and Morrison [29], and Marsden [28]. We shall show, however, that the relativistic version of these theories (either interacting with gravity or with a fixed background) do admit a simple covariant Poisson bracket description. Thus, by means of (1.1) we obtain a unifying principle for media as well as pure fields. For relativistic media field theories written in either Eulerian or Lagrangian description but in 3 + 1 dynamical formulation, a Poisson bracket formulation is known; see Bialynicki-Birula and Iwinski [6], Iwinski and Turski [20], Bialynicki-Birula, Hubbard and Turski [S], Kunzle and Nester [26], Tulczyjew [43], Holm and Kupershmidt [17], Bao, Marsden and Walton [3], and Holm [ 151. We note that the idea of covariant canonid variables has been around for awhile; it is presented, for example, in Barut [4]. Because of its generality, Eq. (1.1) is a natural starting point for obtaining 3 + 1 reductions that result in Hamiltonian formalisms. To see how this works for our motivating particle mechanics example, we suppose that the F of (1.6) has the form
595/1fJ9:1-3
32
MARSDENETAL.
(1.7) where n(t) is an arbitrary function of time and 9 is an arbitrary function of the q’s and p’s. Upon inserting (1.7) and (1.3) into (1.6) we obtain {F, S)(y) = j n(t)@ - (9, H}‘3’) dt = 0,
where (9, H}‘3’ is the conventional
(1.8)
Poisson bracket. Since n(t) is arbitrary,
9 = (9, H)(3).
(1.9)
This is, of course, equivalent to (1.2). Starting with a symplectic formulation of classical field theory for pure fields, Gimmsy [ 141 shows how to obtain the 3 + 1 adjoint Hamiltonian form of ADM and Fischer and Marsden [ 11, 121 (a field theoretic generalization of (1.9) with arbitrary spacetime slicings). The results here give an alternative setting for the same procedures. We illustrate this for electromagnetism in Section 2. The formulation can also be shown to yield the 3 + 1 brackets of Bao, Marsden and Walton [3] for general relativistic fluids. In addition, the incorporation of covariant momentum maps should be possible for these covariant Poisson structures, as well as a covariant version of the reduction procedure (Marsden and Weinstein [33]). The latter would enable one, for example, to pass directly from a covariant Hamiltonian description of a relativistic fluid or plasma in material representation to one in spacetime representation (see Holm [ 151 for some results in this direction). The plan of the paper is as follows. We will first present Maxwell’s equations, the relativistic Maxwell-Vlasov system, general relativity and general relativistic fluids as examples. The covariant Poisson bracket form is exhibited explicitly in each case and the 3 + 1 transition for electromagnetism is given. (Other examples are similar; the authors have treated additional cases, such as the Einstein-Maxwell, Yang-Mills or relativistic Liouville equations. For the non-relativistic Liouville equation, see Marsden, Morrison and Weinstein [30]). We conclude with some remarks on how these results suggest a general formulation of classical field theory. 2. ELECTROMAGNETISM
To begin, we deal with Maxwell’s equations on Minkowsi spacetime. For the usual Euler-Lagrange variational principles, see, for example, Jackson [21]. (The results are generalizable to arbitrary background spacetimes and to general gauge fields.) Let A denote the four vector potential, thought of as a one-form on Minkowski space. Let F = dA,
i.e.,
Fpv = a,A, - d,,A,
(2.1)
COVARIANT
POISSON
BRACKETS
FOR
CLASSICAL
FIELDS
33
be the electromagnetic field tensor, where ap = a/W; p = 0, 1, 2, 3; and x0, x1, x’, x3 are the usual Minkowski coordinates. The standard Lagrangian for the theory with an external current density Jp is L[A]=jY:=j(-+Fp,F”“-A,,J’)d4x
(2.2)
where indices are raised and lowered using the Minkowski metric. In order to define a Legendre transformation, we introduce the covariant momentum variables, 71~”as follows:
The primary constraint manifold is defined to be the image of the map defined by (2.3), where Y is regarded as defined on the space of A, and 8, A,‘s. This image space is the space of pairs of fields (A,, C) with +“ skew symmetric, and is our basic covariant phase space. If F is a functional of A and r, one defines the functional derivatives as usual, being cautious about the constraint on +‘l’ (just as one must be cautious about the div B = 0 constraint in the MHD and Maxwell-Vlasov equations). Namely, 6F/6+” is a skew tensor satisfying
for SrF a skew symmetric perturbation. The covariant Poisson bracket of two functions F and G of A, and 7~~’is defined by (2.4) where v” is an arbitrary vector field on spacetime, and functional derivatives are defined as usual. (The vector field V is related to the passage to dynamical equations-see Remark 3 in Section 6.) The bracket (2.4) can be written as
where
is the associated density,
(2.5)
34
MARSDEN
ET AL.
Let S be defined by the covariant analogue of (1.3), namely
S[A,n] = j [7f”AII*” - H(A,7t)]d‘k,
(2.6)
where H(A, 7c)= $cJc~v + A,JP = &‘“A jl.’ - 9.
(2.7)
We claim that Maxwell’s equations are equivalent
to
(F, S),. (A, n)=O
(2.8)
for all V and F. The statement (2.8) is clearly equivalent iis -0 pF-
and
to
ss
(2.9)
6A,,=O’
i.e., to 52j”‘= -(aPAI’-$A”)
and
$“’
.v
=
-
JP
(2.10)
which, together with F= dA, are the Maxwell equations. (We remark that the choice s(A, n) = j [$T~“‘F,,,, - H(A, n)] d4x would have yielded skew symmetry of rc as one of the consequences of (2.8), but (2.6) seems to be a more useful version for the general theory; in fact one has, in general, a fair amount of freedom in the choice of S. We have followed an analogue of the form (1.3). In other cases, for example gravity, we do not follow such an analogue, but rather directly transcribe the Lagrangian into phase space variables.) Let us now see how this relates to the standard 3 + 1 canonical theory in which A and -E are conjugate variables (see the earlier references or Marsden and Weinstein [34], for example). We choose coordinates so the spacetime vector field V is
To get a closed system, we choose J= 0 and rewrite S from (2.6) as S = j { j [TC’~A,~,- &‘I d?}
dt,
(2.12)
where Latin indices run over 1, 2, 3 and where is? = @,7t” + nio7ti0).
(2.13)
COVARIANT
POISSON
BRACKETS
FOR CLASSICAL
FIELDS
35
Choose F[qS, n] = j” n(t) F[A,,
do] dt,
(2.14)
where F is a function of the 3 + 1 variables Ai and #‘. Clearly ;=n(t); I
I
A dt
(2.15) Hence
(2.16) where I?=
I
Rd3s
(2.17)
and {, j(3) is the usual canonical Poisson bracket for functionals of the canonically conjugate variables A,, rci = rc”. Since n(t) is arbitrary (2.16) yields (2.18) In deriving (2.18) we regard Z? as a function of just Ai and xi. To do this, we reinsert the relation
This is an indication of a general and well-known feature of 3 + l’ing: one must do more than simply replace the rcs with 7cA=x+-the left out momenta rc> must be regarded as functions of the 4” and xA through the Legendre transformation. If we set rci = -E’, ?+ = &OkBk, we thus recover the usual canonical formalism for electromagnetism. (The formalism with E and B as the basic variables-the Pauli-Born-Infeld bracket-requires a reduction by the gauge group of electromagnetism. See Marsden and Weinstein [34]).
36
MARSDEN ETAL. 3. THE RELATIVISTIC MAXWELL-VLASOV
EQUATIONS
A special relativistic particle moves in an external electromagnetic according to the Lorentz force law dx” - up; x--
field F= dA
(3.1)
where z is the particle’s proper time, e is its charge and m its rest mass. Declare pP=mu,+eA
to be canonically
p
(3.2)
conjugate to X~ and set
Thus (3.1) are equivalent to Hamilton’s dxP -=-; dz A relativistic lines:
c
a*
ap,
equations aA dPP = -a* = e uvY. dz axp c axp
(3.4)
plasma density f(x, p) d4xd4p is constant along its particles’ world
df af up +- eaf-
-=dz
axp
cap,
up YdA = 0.
axp
(3.5)
We may rewrite this as (3.6)
The basic field for the Vlasov theory is the plasma phase space density function. As in Iwinski and Turski [20] and in the non-relativistic case (Morrison [37] and Marsden and Weinstein [34]) we define the bracket of two functionals F, G off to be of Lie-Poisson form: (3.7
Let
WI
= {Ax, P) H(T PI &X&P
(3.8
COVARIANT
so SS/Sf = H. An integration
POISSON
BRACKETS
FOR CLASSICAL
37
FIELDS
by parts shows that the covariant bracket equation (3.9)
(6 SHf)=O
is equivalent to the relativistic Vlasov equation (3.5) [or (3.6)]. (As in Kaufman and Holm [22, p. 278), one must suitably restrict the fields and functionals so these integrals converge.) The basic fields for the relativistic Maxwell-Vlasov equations are triples rP”, f ). The bracket of two functions of (A, n, f) is just the sum of (2.3) and
(3.10)
Let S[A,n,f]=I(n”A,.-tn,,.n”)d4x A,)(@
-f
Ap) d4xd“p.
(3.1 : )
The field equations are (3.12)
{F,S}.(A,n,f)=O
for all F and all V. These are obviously equivalent -=6s 0 67c~”
and
to
6S 6A,,=O
and
(3.13) If {$$}~y~d4xd4p=0.
These are, in turn, equivalent to the relativistic CISu”+eiifUv!%=o
axp
cap,
Maxwell-Vlasov
equations
axp
a, Fpv = f s uyf(x, p) d4p E;, = 8, A, - a, A,
(3.14)
38
MARSDENETAL.
Remarks. 1. Here we have not mentioned the obvious physical constraint that f vanishes unless u%, = - 1. This can be treated a posteriori since it can be shown that iff(x, p) is a solution of the relativistic Vlasov equation (3.5) defined on all of xp space, then g(x, p) =f(x, p) S(U%, + 1), where u, = (l/m)(p, - eA./c), also is a solution. Alternatively, this constraint can be treated by restricting to density functions concentrated on the mass shell at the outset.
2. The bracket (3.7) is literally the Lie-Poisson bracket for the group of canonical transformations on xp space, the cotangent bundle of spacetime. Thus, this part of the bracket can be regarded as the reduction from canonical coordinates in Lagrangian representation by the particle relabeling group. In Lagrangian representation, the bracket has a form similar to (2.3); the vector field Y should disappear during reduction because one relabels by world lines, not by points (I, p). This is part of a general covariant reduction process which is planned for future development. 3. Another reduction process that we plan to pusue is the elimination of the gauge freedom for electromagnetism via reduction. This should re-express the bracket in terms of FPv and f alone and build in the div E constraint. When expressed dynamically, this should reproduce the known bracket for relativistic plasmas (Iwinski and Turki [20] and Bialynicki-Birula, Hubbard and Turski [S]), and should coincide with the non-relativistic bracket (Morrison [37], Marsden and Weinstein [ 341).
4. GENERAL RELATIVITY
The basic field variables we use for general relativity are the contravaiant symmetric two-tensor g@ representing the dual metric and the “conjugate momenta” rc$ which are symmetric in c(and /I. We shall identify rr$ with the affine connection; this is standard, although not strictly true from the point of view of the Legendre transformation because of second derivatives of gap in the Lagrangian density (see Misner, Thorne and Wheeler [36, Chap. 211, Kijowski and Szczyrba [24], and Szczyrba [42] ). The Poisson brackets are of the same form as (2.3) namely (4.1)
Here functional derivatives are defined so that those with respect to g”” are tensors:
COVARIANT
POISSON
BRACKETS
FOR
CLASSICAL
FIELDS
39
whereas those with respect to 7~;~are tensor densities: d z
F(rc + 1611) = j” z
hc$d4x.
j.=O
The action is the usual one written in terms of gzp and ?rcD: (4.2)
where ,,&
d4x is the volume element on spacetime, R,, = a,lT;, -
aljnfj,- 7rfp:, + 7cfj,7r;,(
(4.3)
and where
is an (externally imposed) stressxnergy tensor. The covariant bracket equations are for all
{F, S>,,=O
F, V:
i.e., (4.4)
The first equation yields the field equations (4.5)
G,,j = gzTxp, where G,, = R,,{ - ;gXp R”,
while the second equation can be shown to imply
(p 61:~
:= a,(p &Qi) + n;,.,p vG + 7-$g”” Jq - n;,,g”fl&
which implies that n is the Levi-Civita ~if,~= wmg,,,
= 0,
(4.6)
connection: + 66,
- 4gd.
(See Misner, Thorne, and Wheeler [36], Chap. 21, Sect. 21.)
(4.7)
40
MARSDEN ETAL.
We may think of (4.2) as depending parametrically on a set of matter and radiation fields 4” through an additional Lagrangian L*. To couple these fields to the gravitational fields (gap, nip) we need a covariant bracket for the 4. Then the equations
ought to be equivalent
to
and the field equations for the 4. This suggestion is followed in the next section. Our treatment of general relativity is of course a reformulation of the standard Palatini variational principle. One interesting feature of our bracket formulation is that it allows an interesting coupling with media fields, as we shall see. On the negative side, our choice of how to write S in (4.2) is somewhat ad hoc following the Palatini formalism and not as close to the form (1.3) as one might like. Also, the correspondence between the covariant bracket (4.1) and the canonical Dirac-ADM bracket through a 3 + 1 process is suggestive, but it has not been worked out. Difficulties of this or equivalent sorts are common to all the canonical or symplectic treatments of general relativity we know of.
5. GENERAL RELATIVISTIC FLUIDS We consider a perfect adiabatic fluid coupled to gravity; see Misner, Thorne, and Wheeler [36, Chap. 221 for background. One can similarly treat, we presume, plasmas coupled to general relativity (the Maxwell-Einstein-Vlasov system) or charged general relativistic fluids or general relativistic MHD. The basic fluid quantities are the following scalar fields: p = fluid mass-energy per unit rest three volume n = baryon number density per unit rest three volume (T= entropy per unit rest three volume p = pressure in a rest frame s = entropy per baryon p = relativistic inertial mass per unit rest three volume. We have the relations 0 = ns
and
u=p+p.
(5.1)
The equation of state has the form P = dn, 0)
(5.2)
COVARIANT
POISSON
and the pressure is determined
BRACKETS
FOR
CLASSICAL
FIELDS
41
by the Legendre transform ap ap p=n~+aej-yp.
(5.3)
The basic fluid variable are taken to be n, 0, and M, = pu,. Here u’ is the four velocity of the fluid, which satisfies u’u, = - 1, i.e., M”A4, = pL?. This constraint is to be imposed after functional derivatives, i.e., variations are taken. Here indices are raised using the Lorentz dual metric gap. The constraint A, = - 1 can either be imposed directly, as we do, or be viewed as a constraint in the sense of Dirac associated to the gauge symmetry of curve reparametrizations. (The latter requires some work on covariant momentum maps-see Section 6 below.) The fluid brackets are taken to be Lie-Poisson with a structure similar to that in the non-relativistic case (Morrison and Greene [39], Dzyaloshinskii and Volovick [lo] ):
The Lie algebra underlying this Lie-Poisson bracket is a semi-direct product of vector fields and (densities x densities), similar to the nonrelativistic case (see Marsden [27], Holm and Kupershmidt [16] and Marsden ef al. [35]). Here, functional derivatives are defined to be vectors or scalars, not densities: -$,-
,. -0
F(M+dsM)=[$SM,fid4x
d a
i.=o
?
F(n+Mn)={$Sn&d4x
and d z
F(o + ASo) = j ~SO
fi
d4x.
j,=O
We note that the two minus signs in (5.4) are in apparent disagreement with the non-relativistic and 3 + 1 version of the theory (see the above references and Bao, Marsden and Walton [3, Eq. (lC.131). However, when the covariant theory is
42
MARSDEN
ET AL.
decomposed into its 3 + 1 parts, this discrepancy should disappear (for example, when a bracket of vector fields on spacetime is decomposed, the result looks like a semi-direct product bracket, but with a relative sign switch due to the signature ( + + + -) of the spacetime metric; cf. Fischer and Marsden [12, Appendix II].) For the coupled system we use the variables ( $3 n:,, M,, 4 0) and use the bracket (5.4) plus (4.1). For the action we take
-8742
g”“M,
M, + V( n, 0)
fi
d4x,
(5.5)
where R,, is given by (4.3) and Un,o)
= i[p(n,
0) - p(n, a)].
(5.4)
We note that the fluid term in (5.5), when evaluated on the constraint set M”M, = p2 is proportional to the integral of the pressure. The covariant bracket equations are {F, s>,,=o
(5.7)
for all F and I’. Choosing F= F( g, 71)gives rc$ = Levi-Civita
connection of g
and G,, = 87cT,,i, where T,, = PU,U~ +m,pt
(5.8)
as in the previous section. In getting &S,,,,/6gED = 87cTzB,we used the constraint g”“M,M, = - 11’ after taking the variation. Choosing F = F(n ) and F = F(o) gives (nu’)t, = 0
and
(mz);, = 0,
(5.9)
i.e., conservation of baryon number and entropy. (The apparent discrepancy in (5.8) by a factor of 2 is discussed in Bao, Marsden and Walton [3].) Finally, choosing F= F(M,) gives V. T= 0, which, of course, also follows from (5.8) and the Bianchi identity.
COVARIANT
POISSON BRACKETS FOR CLASSICAL FIELDS
6. GENERAL CANONICAL
43
FIELD THEORIES
We sketch here a framework in which the canonical brackets (2.5) and (4.1) can be constructed and in which the Euler-Lagrange equations for a pure field are equivalent to the covariant bracket equations. As we have remarked this also covers, in principle, fluids and plasmas by reduction of this structure from Lagrangian (material) representation to Eulerian (spatial) representation. (In 3 + 1) form, the connection between these is discussed in Holm [15].) General relativity, as usual, is anomalous: it is formally similar, but does not quite fit the scheme presented here. Our fields are assumed to be sections of a vector bundle rr: Y + X over a base manifold X (we take X to be spacetime-but for plasmas it is T*(spacetime) or the mass hyperboloid therein). We suspect that most of what we describe also works for a general fiber bundle, but we have restricted to the vector bundle case for simplicity. The fields are described in local coordinates by dA(xp), where A is a multiindex for field components and .x?’ are spacetime coordinates. Let Y be a given Lagrangian density defined on J’(Y), the first jet bundle of Y. Recall that the fiber .I:( Y) of J’(Y) over a point ?: E T, is J,‘.(Y)= Y,@T~;X=Y(T,X,
Y,).
The Lagrangian density of a field C$is locally given by U(&“, equations are the usual Euler-Lagrange equations for 9,
(6.1)
8,d”).
The field
(6.2)
and we set
We now describe (6.3) intrinsically (cf. Kijowski the bundle of four forms (densities) over X so
and Tulczyjew [25]).
9 : J'( Y) -+ A"(X).
Let A4X be
(6.4)
Let P be the bundle over X whose fiber at x is P,=(Y,0T.~X)*On4,X~TT,XOY:o/14,X. Describe P by local coordinates derivative of Y:
(@, z;). The Legendre transformation
(6.5)
is the fiber
44
MARSDEN
ET AL.
given locally by
where n; is given by (6.3). Let F and G be functionals of sections of P. Then we have a Poisson bracket
where ~F/c%$~ is a section of Y* 0 A4X + X and 6G/6n$ is a section of Y@ T*X-+ X. They pair together to give a section of T*X@ A4X + X which can be contracted with the vector field V and the resulting four form integrated over X. For each fixed V, the bracket makes the sections of P into a Poisson manifold. The primary constraint set C is the image of the Legendre transformation. We will assume that it is a vector subbundle of P. This will be the case, for example, if 2’ is quadratic in 8,4” and if its “kinetic matrix” 8’2/a(a,4”) 8(8,@) has constant rank. Let 1: P + P be a smooth vector bundle projection with im I = C. For example, in electromagnetism, 1 would project any tensor density onto its skew-symmetric part. Then l*: Y@ T*X + Y@ T*X. Set C* = im I*, a subbundle of Y@ T*x. This is a bundle dual to C, so that functional derivatives with respect to the constrained covariant momenta naturally take values in C*:
where F is a functional of sections of C, &r is a variation in C and c?F/Sz~ takes values in C*; thus (6F/h$). SK: is a density on X. One may now define brackets of functionals on T(C) (sections of C) by the same formula as before (6.6), however, where the 8/&r: are interpreted as sections of C*. These brackets satisfy all the conditions for Poisson brackets. The only non-obvious condition is the Jacobi identity. To check this, we extend functionals F on f(C) to functionals F= i*F on r(P) as follows: %A xl= F(hln). For a general extension r’, we have
by the fiber linearity of 1. However, for our extension,
COVARIANT
F’OISSON
BRACKETS
FOR
CLASSICAL
FIELDS
45
since /y(S&%r) = 0, where I, = id - 1. Note that the bracket of two such extensions is again such an extension and hence {F,iC}={F,G}-.
(6.7)
Now Jacobi’s identity follows since it holds for the extended functions. Remarks. 1. Equation (6.7) is expressed by saying that (4, X) I-+ (4, /7~) is a Poisson map. Thus the constrained brackets are the pull back of the full brackets by the projection T(P) + r(C). Note that the injection F(C) + r(P) is not a Poisson map for C # P. In fact,
for arbitrary extensions. 2. For general fiber bundles, or if C is not a vector bundle, the results just described require extension. This development should be done in conjunction with examples such as general relativistic fluids written in the Lagrangian (material) picture. The “Hamiltonian
density” is uniquely defined on C by
(6.7) H(4, n) = WrdA - w$A, Q”). At first, the right-hand side is defined on J’(Y) x C. However, the partial derivative with respect to a,$” is zero, so we get a well-defined density on C. Set
and note that
6S
and
jg=O,
(6.9)
for all F, V
(6.10)
i.e., (F, S} “=O reproduce respectively 6H
a 8XP -nn:,=
which are equivalent Remarks. 1. In canonical brackets, canonical brackets covariant version [31, 321.
-v
to the Euler-Lagrange
and
(6.11) equations (6.2).
the above setting, only canonical brackets are described. Nonsuch as those for fluids and plasmas, are expected to come from in Lagrangian representation as in the non-relativistic case by a of the reduction process. See Marsden, Ratiu and Weinstein
46
MARSDEN ETAL.
2. We conjecture that covariant momentum maps associated with a group action G on P should be defined to be maps J: P --f g* Q TXQ A4P where 9 is the Lie algebra of G. These should be consistent with the covariant momentum maps defined in Gimmsy Cl43 and should include standard Noether identities. As in Gimmsy [14], one can presumably show that for an appropriately covariant localized theory, J vanishes on solutions of the field equations and that these conditions J= 0 correspond to first class constraints in the sense of Dirac [9]. The momentum maps should play a key role in the reduction process, as in the nonrelativistic case (Marsden et al. [35]. 3. As already noted, the mathematical development of a systematic 3 + 1 analysis is incomplete. This requires further development of the theory along the lines of Remarks 1 and 2. Once this is done, the 3 + 1 analysis should proceed as in Gimmsy [ 141. In particular, the 3 + 1 procedure applied to the covariant brackets and field equations should directly yield the dynamical Poisson brackets and the evolution equations in bracket form (which is equivalent to the adjoint form of Fischer and Marsden [ 11, 123). As we saw in Section 2, the vector V in the bracket (6.6) plays an important role in the 3 + 1 process. It corresponds to the arbitrariness in the choice of the direction of time and to the lapse and shift which appear in the dynamical formulation. Forming the variables 4” and nA = ~5 V, is a first step in constructing conjugate variables for the 3 + 1 formalism. Subsequently, one must also eliminate the so-called “atlas fields” (such as the temporal component of A in electromagnetism), as in Gimmsy [ 141. 4. The results of this paper also need to be studied with a view towards understanding limits and averaging (see, for example, Weinstein [44] and Similon, Kaufman and Holm [40]). For example, we presume that the fluid bracket (5.4) can be derived from the plasma bracket (3.10) in the cold plasma limit (Gibbons, Holm and Kuperschmidt [13]) and that, as in Marsden et al. [35], taking moments via reduction gives a Poisson map between these structures.
ACKNOWLEDGMENTS We are grateful remarks.
to D. Bao, G. Gimmsy,
D. Helm,
A. Kaufman
and T. Ratiu
for helpful
discussions
and
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FIELDS
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